Embedded newform 72.4.l.b.11.31

• Label: 72.4.l.b.11.31
• Level: $72$
• Weight: $4$
• Character: 72.11
• Analytic conductor: $4.248$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24813752041\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.31
Character	\(\chi\)	\(=\)	72.11
Dual form			72.4.l.b.59.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.82386 + 0.160583i) q^{2} +(-5.18814 - 0.288534i) q^{3} +(7.94843 + 0.906929i) q^{4} +(5.34568 + 9.25900i) q^{5} +(-14.6043 - 1.64791i) q^{6} +(15.5169 + 8.95867i) q^{7} +(22.2996 + 3.83743i) q^{8} +(26.8335 + 2.99391i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 3 q^{2} + 6 q^{3} - 17 q^{4} - 3 q^{6} + 42 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.82386	+	0.160583i	0.998387	+	0.0567746i
							\(3\)	−5.18814	−	0.288534i	−0.998457	−	0.0555285i
\(5\)	5.34568	+	9.25900i	0.478132	+	0.828150i								0.999686	−	0.0250690i	\(-0.00798055\pi\)
							−0.521553	+	0.853219i	\(0.674647\pi\)
\(7\)	15.5169	+	8.95867i	0.837832	+	0.483723i	0.856527	−	0.516103i	\(-0.172618\pi\)
							−0.0186945	+	0.999825i	\(0.505951\pi\)
\(11\)	−19.0639	−	11.0065i	−0.522543	−	0.301690i	0.215432	−	0.976519i	\(-0.430884\pi\)
							−0.737974	+	0.674829i	\(0.764217\pi\)
\(13\)	17.9061	−	10.3381i	0.382021	−	0.220560i	−0.296676	−	0.954978i	\(-0.595878\pi\)
							0.678697	+	0.734418i	\(0.262545\pi\)
\(17\)			36.2107i			0.516611i	0.966063	+	0.258306i	\(0.0831642\pi\)
							−0.966063	+	0.258306i	\(0.916836\pi\)
\(19\)	−125.564			−1.51612			−0.758062	−	0.652182i	\(-0.773854\pi\)
							−0.758062	+	0.652182i	\(0.773854\pi\)
\(23\)	−88.1103	−	152.612i	−0.798794	−	1.38355i	−0.920402	−	0.390974i	\(-0.872138\pi\)
							0.121607	−	0.992578i	\(-0.461195\pi\)
\(29\)	−77.3286	+	133.937i	−0.495157	+	0.857638i	−0.999984	−	0.00558276i	\(-0.998223\pi\)
							0.504827	+	0.863221i	\(0.331556\pi\)
\(31\)	243.700	−	140.701i	1.41193	−	0.815179i	0.416362	−	0.909199i	\(-0.363305\pi\)
							0.995570	+	0.0940195i	\(0.0299716\pi\)
\(37\)		−	223.656i		−	0.993754i	−0.867821	−	0.496877i	\(-0.834480\pi\)
							0.867821	−	0.496877i	\(-0.165520\pi\)
\(41\)	47.3067	−	27.3125i	0.180197	−	0.104037i	−0.407188	−	0.913344i	\(-0.633491\pi\)
							0.587385	+	0.809308i	\(0.300157\pi\)
\(43\)	157.422	−	272.663i	0.558294	−	0.966994i	−0.439345	−	0.898319i	\(-0.644789\pi\)
							0.997639	−	0.0686756i	\(-0.0218774\pi\)
\(47\)	−230.270	+	398.839i	−0.714644	+	1.23780i	0.248453	+	0.968644i	\(0.420078\pi\)
							−0.963097	+	0.269156i	\(0.913255\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.4.l.b.11.31	yes	64
3.2	odd	2		216.4.l.b.35.2		64
4.3	odd	2		288.4.p.b.47.32		64
8.3	odd	2	inner	72.4.l.b.11.23	✓	64
8.5	even	2		288.4.p.b.47.31		64
9.4	even	3		216.4.l.b.179.10		64
9.5	odd	6	inner	72.4.l.b.59.23	yes	64
12.11	even	2		864.4.p.b.143.8		64
24.5	odd	2		864.4.p.b.143.25		64
24.11	even	2		216.4.l.b.35.10		64
36.23	even	6		288.4.p.b.239.31		64
36.31	odd	6		864.4.p.b.719.25		64
72.5	odd	6		288.4.p.b.239.32		64
72.13	even	6		864.4.p.b.719.8		64
72.59	even	6	inner	72.4.l.b.59.31	yes	64
72.67	odd	6		216.4.l.b.179.2		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.l.b.11.23	✓	64	8.3	odd	2	inner
72.4.l.b.11.31	yes	64	1.1	even	1	trivial
72.4.l.b.59.23	yes	64	9.5	odd	6	inner
72.4.l.b.59.31	yes	64	72.59	even	6	inner
216.4.l.b.35.2		64	3.2	odd	2
216.4.l.b.35.10		64	24.11	even	2
216.4.l.b.179.2		64	72.67	odd	6
216.4.l.b.179.10		64	9.4	even	3
288.4.p.b.47.31		64	8.5	even	2
288.4.p.b.47.32		64	4.3	odd	2
288.4.p.b.239.31		64	36.23	even	6
288.4.p.b.239.32		64	72.5	odd	6
864.4.p.b.143.8		64	12.11	even	2
864.4.p.b.143.25		64	24.5	odd	2
864.4.p.b.719.8		64	72.13	even	6
864.4.p.b.719.25		64	36.31	odd	6